A game in extensive form (with perfect information and chance moves) is a vector

= (N, V, E, x0,(Vi)i∈N∪{0},(px)x∈V0, O, u), (3.6) where:

r N is a finite set of players.

r (V , E, x0) is the game tree.

r (Vi)i∈N∪{0}is a partition of the set of vertices that are not leaves.

r For every vertex x∈ V0, px is a probability distribution over the edges emanating

51 3.5 Games with chance moves

rO is the set of possible outcomes.

ru is a function mapping each leaf of the game tree to an outcome in O.

The notation used in the extension of the model is the same as the previous notation, with the following changes:

rThe partition of the set of vertices is now (Vi)i∈N∪{0}. We have, therefore, added the set V0

to the partition, where V0is the set of vertices at which a chance move is implemented. rFor each vertex x ∈ V0, a vector px, which is a probability distribution over the edges

emanating from x, has been added to the model.

Games with chance moves are played similarly to games without chance moves, the only difference being that at vertices with chance moves a lottery is implemented, to determine the action to be undertaken at that vertex. We can regard a vertex x with a chance move as a roulette wheel, with the area of the pockets of the roulette wheel proportional to the values px. When the game is at a chance vertex, the wheel is spun, and the pocket at which

the wheel settles specifies the new state of the game.

Note that in this description we have included a hidden assumption, namely, that the probabilities of the chance moves are known to all the players, even when the game includes moves that involve the probability of rain, or an earthquake, or a stock market crash, and so forth. In such situations, we presume that the probability assessments of these occurrences are known by all the players. More advanced models take into account the possibility that the players do not all necessarily share the same assessments of the probabilities of chance moves. Such models are considered in Chapters 9, 10, and 11.

In a game without chance moves, a strategy vector determines a unique play of the game (and therefore also a unique game outcome). When a game includes chance moves, a strategy vector determines a probability distribution over the possible game outcomes.

Example 3.18 (Continued ) (See Figure3.8.) Suppose that Player I uses strategy sI, defined as

sI(R)= b, sI(C)= h, (3.7)

and that Player II uses strategy sII, defined as

sII(B)= f. (3.8)

Then:

rthe play R→ A → B → (2, 0) occurs with probability 1/2, leading to outcome (2, 0);

rthe play R→ A → C → E → (0, 2) occurs with probability 1/8, leading to outcome (0, 2);

rthe play R→ A → C → E → (−1, 1) occurs with probability 1/4, leading to outcome (−1, 1);

rthe play R→ A → C → E → (1, 1) occurs with probability 1/8, leading to

outcome (1, 1). _

Using this model of games with chance moves, we can represent games such as backgammon, Monopoly, Chutes and Ladders, and dice games (but not card games such as poker and bridge, which are not games with perfect information, because players do

not know what cards the other players are holding). Note that von Neumann’s Theorem (Theorem 3.13) does not hold in games with chance moves. In dice games, such as backgammon, a player who benefits from favorable rolls of the dice can win regardless of whether or not he has the first move, and regardless of the strategy adopted by his opponent.

3.6

Games with imperfect information

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • One of the distinguishing properties of the games we have seen so far is that at every stage of the game each of the players has perfect knowledge of all the developments in the game prior to that stage: he knows exactly which actions were taken by all the other players, and if there were chance moves, he knows what the results of the chance moves were. In other words, every player, when it is his turn to take an action, knows precisely at which vertex in the game tree the game is currently at. A game satisfying this condition is called a game with perfect information.

The assumption of perfect information is clearly a very restrictive assumption, limiting the potential scope of analysis. Players often do not know all the actions taken by the other players and/or the results of chance moves (for example, in many card games the hand of cards each player holds is not known to the other players). The following game is perhaps the simplest example of a game with imperfect information.

Example 3.20 Matching Pennies The game Matching Pennies is a two-player game in which each player chooses one of the sides of a coin, H (for heads) or T (for tails) in the following way: each player inserts into an envelope a slip of paper on which his choice is written. The envelopes are sealed and submitted to a referee. If both players have selected the same side of the coin, Player II pays one dollar to Player I. If they have selected opposite sides of the coin, Player I pays one dollar to Player II. The depiction of Matching Pennies as an extensive-form game appears in Figure3.9. In Figure3.9, Player I’s actions are denoted by upper case letters, and Player II’s actions are depicted by lower case letters.

I R A B H T h t h t (1, −1) (−1, 1) (1, −1) (−1, 1) II

Figure 3.9 The game Matching Pennies as a game in extensive form

Figure3.9introduces a new element to the depictions of extensive-form games: the two vertices Aand B of Player II are surrounded by an ellipse. This visual element represents the fact that when Player II is in the position of selecting between h and t, he does not know whether the game state is currently at vertex A or vertex B, because he does not know whether Player I has selected H or T. These two vertices together form an information set of Player II. 

53 3.6 Games with imperfect information

Remark 3.21 The verbal description of Matching Pennies is symmetric between the two